the nonlinguistic real world.
Metalanguage is used to describe language, either object language or metalanguage.
The language of science is typically expressed in the object-language perspective, while much of the discourse in philosophy of science is in the metalinguistic perspective. Terms such as “theory” and “explanation” are examples of expressions in metalanguage.
3.03 Dimensions of Language
The metalinguistic perspective includes what may be called dimensions of language, which serve well as an organizing framework for philosophy of language. Four dimensions may be distinguished for philosophy of language. They are A. syntax, B. semantics, C. ontology and D. pragmatics. Most philosophers of science ignore the linguists’ phonetic and phonemic dimensions. And most linguists ignore ontology.
A. SYNTAX
3.04 Syntactical Dimension
Syntax is the system of linguistic symbols considered in abstraction from their associated meanings.
Syntax is the most obvious part of language. It is residual after abstraction from pragmatics, ontology, and semantics. And it consists only of the forms of expressions, so it is often said to be “formal”. Since meanings are excluded from the syntactical dimension, the expressions are also said to be semantically “uninterpreted”. And since the language of science is usually written, the syntax of interest consists of visible marks on paper or more recently linguistic displays on computer display screens the syntax of expressions is sometimes called “inscriptions”. Examples of syntax include the sentence structures of colloquial discourse, the formulas of pure or formal mathematics, and computer source codes such as FORTRAN or LISP.
3.05 Syntactical Rules
Syntax is a system of symbols. Therefore in addition to the syntactical symbols and structures, there are also rules for the system called “syntactical rules”. These rules are of two types: formation rules and transformation rules.
Formation rules are procedures described in metalanguage that regulate the construction of grammatical expressions out of more elementary symbols.
Formation rules order such syntactical elements as mathematical variables and operator signs, descriptive and syncategorematic terms, and the user-defined variable names and reserved words of computer source codes. Expressions constructed from the symbols in compliance with the formation rules for a language are called “grammatical” or “well formed formulas”, and include the computer instructions called “compiler-acceptable” and “interpreter-acceptable” source code.
When there exists an explicit and adequate set of syntactical formation rules, it is possible to develop a type of computer program called a “mechanized generative grammar”. A generative grammar constructs grammatical expressions from inputs consisting of more elementary syntactical symbols. The generative-grammar computer programs input, process, and output object language, while the source-code instructions constituting the computer system are therefore metalinguistic expressions.
A mechanized generative grammar is a computer system that applies formation rules to more elementary syntactical symbols inputted to the system, and thereby outputs grammatically well formed expressions.
When a mechanized generative grammar is used to produce new scientific theories in the object language of a science, the computer system is called a “discovery system”. Typically the system also contains an empirical test criterion for the selection of a subset for output of the numerous theories generated.
A discovery system is a mechanized generative grammar that constructs and may also empirically test scientific theories as its output.
Transformation rules change grammatical sentences into other grammatical sentences.
For example there are transformation rules for colloquial discourse that change a sentence from declarative to interrogative mood. The object language of science is typically expository, and philosophy of science therefore principally considers the declarative mood for the descriptive discourse as in theories and laws. The imperative mood is also of interest for describing procedural instructions in test designs for executing the tests.
Transformation rules are used in logical and mathematical deductions. But logic and mathematical rules are intended not only to produce new grammatical sentences but also to guarantee truth transferability from one set of sentences or equations to another to generate theorems, usually by the transformation rule of substitution that makes logic extensional.
In 1956 Herbert Simon developed an artificial-intelligence computer system named LOGIC THEORIST, which operated with his “heuristic-search” system design. This system developed deductive proofs of the theorems in Alfred N. Whitehead and Bertrand Russell’s Principia Mathematica. The symbolic-logic formulas are object language for this system. But Simon correctly denies that the Russellian symbolic logic is an effective metalanguage for the design of discovery systems.
Transformation rules are of greater interest to linguists, logicians and mathematicians than to contemporary philosophers of science, who recently have been more interested in formation rules for generative-grammar discovery systems.
3.06 Mathematical Language
The syntactical dimension of mathematical language includes mathematical symbols and the formation and transformation rules of the various branches of mathematics. Mathematics applied in science is object language for which the syntax is supplied by the mathematical formalism. Whenever possible the object language of science is mathematical rather than colloquial, because measurement values for variables enable the scientist to quantify the error in his theory, after estimates are made for the range of measurement error, usually by repeated execution of the measurement procedure.
3.07 Logical Quantification in Mathematics
Mathematical expressions in science are universally quantified when descriptive variables have no associated numerical values, and are particularly quantified when numeric values are associated with any of the expression’s descriptive variables either by measurement or by calculation.
Like categorical statements, mathematical equations are explicitly quantified logically as either universal or particular, even though the explicit indication is not by means of the syncategorematic logical quantifiers “every”, “some” or “no”. An equation in science is universally quantified logically when none of its descriptive variables are assigned numeric values. Universally quantified equations may contain mathematical constants as in some theories or laws. An equation is particularly quantified logically by associating measurement values with any of its descriptive variables. A variable may then be said to describe an individual measurement instance.
When a numeric value is associated with a descriptive variable by computation with measurement values associated with other descriptive variables in the same mathematical expression, the variable’s calculated value may be said to describe an individual empirical instance. In this case the referenced instance has not been measured but depends on measurements associated with other variables in the same equation.
Individual empirical instances are calculated when an equation is used to make a numerical prediction. The individual empirical instance is the predicted value, which makes an empirical claim. In a test it is compared with an individual measurement
